Means and variances for a family of similarity indices used in cluster analysis

Albatineh et al. (2006) introduced a family LL of similarity indices. Members of this family are linear functions of the matching counts matrix [mij], where mij is the number of common elements between the i th and j th clusters resulting from two clusterings of the same data set. Fowlkes and Mallows (1983) derived the mean and variance for Rand (1971) index and an index they called Bk (which is actually attributed to Ochiai, 1957) under fixed marginal totals of the matching counts matrix and independence of the clustering algorithms. This paper generalizes the derivation of Fowlkes and Mallows (1983) for the mean and variance to any member of the LL family which makes the problem of comparison of a wide family of indices much easier. Monte Carlo simulations are implemented to compare shapes, means and variances for nine members of the LL family for null case data (without clustering structure). Structured case simulations are implemented to evaluate the nine indices as tools for measuring cluster structure recovery. Data were generated from bivariate normal distributions.